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Strong and Weak Identifiability of Optimization-based Causal Discovery in Non-linear Additive Noise Models

ICML 2025 · Probabilistic and scientific modeling › Causal discovery · approach: Constraint-based structure learning with latents

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Defines strongly vs. weakly identifiable ANM causal structures via implicit functions and proposes GENE, an order-based causal discovery method that combines R² goodness-of-fit with residual independence tests and a least-pruning step.

Problem

Optimization-based causal discovery under the additive noise model performs well on some problems but poorly on others; identifiability difficulty varies even within ANM with the same graph structure.

Approach

Classifies identifiable ANM structural equations as strongly or weakly identifiable using an implicit-function condition, and proposes an order-search fitness combining R² goodness of fit (scale invariant) with chi-square residual-independence penalties so that one method handles both regimes.

Key results

• On the real Sachs dataset GENE achieves F1 0.65 and SHD 11, versus 0.38/16 for the next best (CAM).
• On weakly identifiable MLP problems, GENE's order Rev is roughly half that of CAM and a quarter of RESIT; on strongly identifiable problems Rev is near 0.
• Removing the residual independence term drops F1 significantly on weakly identifiable MLP data while leaving strongly identifiable MIM and GP performance largely unaffected.

Measured effects reported in the paper

Surprising findings

• Continuous-optimization methods (Notears-MLP, Gran-DAG) lose F1 heavily after standardization, with a 52% average decline on weakly identifiable problems versus 34% on strong ones.
• Residual independence also helped GENE find true orders in strongly identifiable settings at high graph density, contrary to expectation.

Limitations

• Pruning strategy performance can be further improved, as the authors note.
• Theory covers ANM only; extension to post-nonlinear and other settings left open.
• Evaluation limited to synthetic ER graphs with n=3000 and one real dataset (Sachs, 853 observational samples).

Experimental setup

Datasets: Sachs synthetic ER graphs (MIM, MLP, GP nonlinearities)

Benchmarks: Sachs synthetic ER graphs (MIM, MLP, GP)

Metrics: F1 Score SHD degree of reverse (Rev) wall-clock time

Baselines: Notears-MLP Gran-DAG CAM RESIT PC-ANM GSF R2Sort CaPS NHTS SELF

Builds on: additive noise model (ANM) CAM RESIT PC-ANM NOTEARS

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